Electrostatic Energy and Capacitance Chapter 26 1 30
Energy of a Charge Distribution How much energy ( work) is required to assemble a charge distribution ?. CASE I: Two Charges Bringing the first charge does not require energy ( work)
Energy of a Charge Distribution How much energy ( work) is required to assemble a charge distribution ?. CASE I: Two Charges Bringing the first charge does not require energy ( work) r Q1 Q2 Bringing the second charge requires to perform work against the field of the first charge.
Energy of a Charge Distribution CASE I: Two Charges r Q1 Q2 Bringing the second charge requires to perform work against the field of the first charge. W = Q2 V1 with V1 = (1/40) (Q1/r) W = (1/40) (Q1 Q2 /r) = U U = potential energy of two point charges U = (1/40) (Q1 Q2 /r)
Energy of a Charge Distribution CASE II: Several Charges How much energy is stored in this square charge distribution?, or … What is the electrostatic potential energy of the How much work is needed to assemble this charge distribution? Q Q Q Q a To answer it is necessary to add up the potential energy of each pair of charges U = Uij U12 = potential energy of a pair of point charges U12 = (1/40) (Q1 Q2 /r)
Energy of a Charge Distribution -Q +Q A fields cancel CASE III: Parallel Plate Capacitor fields add E d fields cancel Electric Field E = / 0 = Q / 0 A ( = Q / A) Potential Difference V = E d = Q d / 0 A 9 32
Energy of a Charge Distribution -Q +Q A fields cancel CASE III: Parallel Plate Capacitor fields add E d fields cancel Now, suppose moving an additional very small positive charge dq from the negative to the positive plate. We need to do work. How much work? dW = V dq = (q d / 0 A) dq We can use this expression to calculate the total work needed to charge the plates to Q, -Q 9 32
Energy of a Charge Distribution -Q +Q A fields cancel CASE III: Parallel Plate Capacitor fields add E d fields cancel dW = V dq = (q d / 0 A) dq The total work needed to charge the plates to Q, -Q, is given by: W = dW = (q d / 0 A) dq = (d / 0 A) q dq W = (d / 0 A) [Q2 / 2] = d Q2 / 2 0 A 9 32
Energy of a Charge Distribution -Q +Q A fields cancel CASE III: Parallel Plate Capacitor fields add E d fields cancel The work done in charging the plates ends up as stored potential energy of the final charge distribution W = U = d Q2 / 2 0 A Where is the energy stored ? The energy is stored in the electric field 9 32
Energy of a Charge Distribution -Q +Q A fields cancel CASE III: Parallel Plate Capacitor fields add E d fields cancel The energy U is stored in the field, in the region between the plates. E = Q / (0 A) U = d Q2 / 2 0 A = (1/2) 0 E2 A d The volume of this region is Vol = A d, so we can define the energy density uE as: uE = U / A d = (1/2) 0 E2 9 32
Energy of a Charge Distribution Electric Energy Density CASE IV: Arbitrary Charge Distribution uE = U / A d = (1/2) 0 E2 Although we derived this expression for the uniform field of a parallel plate capacitor, this is a universal expression valid for any electric field. When we have an arbitrary charge distribution, we can use uE to calculate the stored energy U . dU = uE d(Vol) = (1/2) 0 E2 d(Vol) U = (1/2) 0 E2 d(Vol) [The integral covers the entire region in which the field E exists] 9 32
A Shrinking Sphere A sphere of radius R1 carries a total charge Q distributed evenly over its surface. How much work does it take to shrink the sphere to a smaller radius R2 ?. 10 40
Capacitance -Q A d +Q C = Q / V [Units: Coulomb /Volt = Farad] Two parallel plates charged Q and –Q respectively constitute a capacitor -Q +Q A E d The electric field between the plates is E = Q / A 0 The potential difference between the plates is V = E d = Q d / A eo The charge Q gives rise to a potential difference V such that V = Q d / A 0 The ratio C = Q / V = A 0 / d is called the capacitance C = Q / V [Units: Coulomb /Volt = Farad] 9 34
Capacitance -Q A d +Q C = Q / V Two parallel plates charged Q and –Q respectively constitute a capacitor -Q +Q A E d C = Q / V The relationship C = Q / V is valid for any charge configuration (Indeed this is the definition of capacitance or electric capacity) In the particular case of a parallel plate capacitor C = Q / V = 0 A / d The capacitance is directly proportional to the area of the plates and inversely proportional to the separation between the plates 9 34
What Does a Capacitor Do? Stores electrical charge. Stores electrical energy. Capacitors are basic elements of electrical circuits both macroscopic (as discrete elements) and microscopic (as parts of integrated circuits). Capacitors are used when a sudden release of energy is needed (such as in a photographic flash). Electrodes with capacitor-like configurations are used to control charged particle beams (ions, electrons).
What Does a Capacitor Do? Stores electrical charge. Stores electrical energy. The charge is easy to see. If a certain potential, V, is applied to a capacitor C, it must store a charge Q=CV: -Q +Q C (Symbol for a capacitor) V
What Does a Capacitor Do? Stores electrical charge. Stores electrical energy. An electrical field is created between the plates an energy U = (1/2) 0 E2 A d = d Q2 / 2 0 A is stored between the plates of the capacitor -Q +Q C V - +
Energy Stored in a Capacitor U = (1/2) 0 E2 A d = d Q2 / 2 0 A But C = 0 A / d Then: U = (1/2) Q2 / C = (1/2) C V2 Potential Energy And: u = (1/2) 0 E2 Potential Energy Density Although we calculated U and u for the parallel plate capacitor, the expressions obtained are valid for any geometry. 10 40
Energy Stored in a Capacitor Suppose we have a capacitor with charge q (+ and -). Then we increase the charge by dq (+ and -). We must do work dW = Vdq to increase charge: 10 40
Energy Stored in a Capacitor Suppose we have a capacitor with charge q (+ and -). Then we increase the charge by dq (+ and -). We must do work dW = Vdq to increase charge: 10 40
Energy Stored in a Capacitor Suppose we have a capacitor with charge q (+ and -). Then we increase the charge by dq (+ and -). We must do work dW = Vdq to increase charge: Integrating q from 0 to Q, we can find the total stored (potential) electric energy: 10 40
Energy Stored in a Capacitor Suppose we have a capacitor with charge q (+ and -). Then we increase the charge by dq (+ and -). We must do work dW = Vdq to increase charge: Integrating q from 0 to Q, we can find the total stored (potential) electric energy: 10 40
Energy Density. +Q -Q V Now compute the energy density, uE, inside the capacitor. For a parallel plate capacitor of volume A.d, uE = U/(Ad) = (1/2 CV2)/Ad But for a parallel plate capacitor , C = e0A/d 11 43
Energy Density. +Q -Q V Now compute the energy density, uE, inside the capacitor. For a parallel plate capacitor of volume A.d, u= U/(Ad) = (1/2 CV2)/Ad But for a parallel plate capacitor , C = e0A/d uE = (e0/2)(V/d)2 = (e0/ 2)E2 11 44
Energy Density. +Q -Q V uE = (e0/2)(V/d)2 = (e0/ 2)E2 This leads to another understanding of electric field The energy is stored in the FIELD, rather than in the plates! If an electric field exists, then you can associate an electric potential energy of (e0/2)E2 11 45
Cylindrical Capacitor Construct Gaussian surface around inner cylinder. Then q = e0EA = e0E (2prL) Therefore E = q/(e02prL) + inner rad. a outer rad. b length L 13 46
Cylindrical Capacitor Construct Gaussian surface around inner cylinder. Then q = e0EA = e0E (2prL) Therefore E = q/(e02prL) + inner rad. a outer rad. b length L 13 47
Cylindrical Capacitor Construct Gaussian surface around inner cylinder. Then q = e0EA = e0E (2prL) Therefore E = q/(e02prL) + inner rad. a outer rad. b length L in Farads 13 48
Cylindrical Capacitor Construct Gaussian surface around inner cylinder. Then q = e0EA = e0E (2prL) Therefore E = q/(e02prL) + inner rad. a outer rad. b length L Note: C is made larger by making (b - a) as small as possible. in Farads 13 49
Energy Stored in a Cylindrical Capacitor The energy stored in a capacitor C (with charge Q and voltage difference V) is given by: Using C = 20 L / ln (b/a) we obtain: U = q2 ln (b/a) / 4 0 L The energy U is stored in the electric field between the two conductors
Parallel and Series Series Parallel
Capacitors in Circuits -Q +Q (Symbol for a capacitor) C V A piece of metal in equilibrium has a constant value of potential. Thus, the potential of a plate and attached wire is the same. The potential difference between the ends of the wires is V, the same as the potential difference between the plates.
Capacitors in Parallel C1 - q1 Suppose there is a potential difference V between a and b. Then q1 V = C1 & q2 V = C2 a b C2 - q2 We want to replace C1 and C2 with an equivalent capacitance C = q V The charge on C is q = q1 + q2 Then C = q V = (q1 + q2 ) V = q1 V + q2 V = C1 + C2 V a b C - q C = C1 + C2 This is the equation for capacitors in parallel. Increasing the number of capacitors increases the capacitance.
Capacitors in Series C1 C2 C a -q +q -q +q b a -q +q b V1 V2 V Here the total potential difference between a and b is V = V1 + V2 Also V1 = (1/C1) q and V2 = (1/C2) q The charge on every plate (C1 and C2) must be the same (in magnitude) Then: V = V1 + V2 = q / C1 +q / C2 = [(1/C1) + (1/C2)] q or, V = (1/C) q 1 / C = 1 / C1 + 1 / C2 This is the equation for capacitors in series. Increasing the number of capacitors decreases the capacitance.
Dielectrics in Capacitors Suppose we fill the space between the plates of a capacitor with an insulating material (a “dielectric”): The material will be “polarized” - electrons are pulled away from atom cores Consequently the E field within the capacitor will be reduced - + +Q -Q - +
Dielectrics in Capacitors Gaussian surface - + - + -q +q -q’ -q +q +q’ - + +q -q E0 - E + We calculate the new field E using Gauss’ Law, and noting there is an induced charge q’ on the surface of the dielectric. E0 = q / (0 A) and E = (q – q’) / (0 A) E0 / E = q / (q – q’) E0 / E = or E = E0/k The field is reduced by factor k E = E0/k The constant is called the Dielectric Constant = q / (q –q’) = 1 / [1-(q’/q)] > 1
Effect on Capacitance A dielectric reduces the electric field by a factor k (E=E0/) Hence V = E d is reduced by k [V = (E0/) d = V0/)] and C= Q/V is increased by k [C = (Q k) / V0= C0 ) parallel plate capacitor with dielectric. Adding a dielectric increases the capacitance.
Dielectrics & Gauss’s Law With a dielectric present, Gauss’s Law can be rewritten from +q -q’ to Instead of having to think about the confusing induced charge q’, we can simply use the free charge q. But E is replaced by keE.